Q. 44 - Calculus

Question

Consider the region between f(x) = x+1 and the line y = −2 on [1, 5]. For each line of rotation given in Exercises 43 and 44, use the shell method to construct definite integrals to find the volume of the resulting solid  

Step-by-Step Solution

Answer

The integral can be given as $2\pi {\int }_1^5(x^2+3x)dx$ the value of the integral is $\frac{464\pi }{3} cubicunits$

Step 1: Given information

We are given a function f(x)= x+1 and

Step 2: Find the integral and evaluate it

We know that integral can be given as ${\int }_a^{b}r\left(x\right)h\left(x\right)dx$ where r and h are the radius and height of the function

the axis of rotation is y-axis hence the radius is $r\left(x\right)=x$

also the height can be given as $h\left(x\right)=x+3$

substituting the values in the integral 

$$\begin{gathered} V=2\pi {\int }_1^{5}x(x+3)dx \\ V=2\pi {\int }_1^5(x^2+3x)dx \\ V=2\pi {{[\frac{x^3}{3}+\frac{3}{2}{x}^2]}^5}_1 \\ V=\frac{464}{3}\pi \end{gathered}$$